Combinatorial sums and implicit Riordan arrays
نویسندگان
چکیده
In this paper we present the theory of implicit Riordan arrays, that is, Riordan arrays which require the application of the Lagrange Inversion Formula to be dealt with. We show several examples in which our approach gives explicit results, both in finding closed expressions for sums and, especially, in solving classes of combinatorial sum inversions.
منابع مشابه
An algorithm for proving identities with Riordan transformations
The problem we consider in the present paper is how to find the closed form of a class of combinatorial sums, if it exists. The problem is well known in the literature, and is as old as Combinatorial Analysis is, since we can go back at least to Euler’s time. More recently, Riordan has tried to give a general approach to the subject, proposing a variety of methods, many of which are related to ...
متن کاملSome Properties of the (p, q)-Fibonacci and (p, q)-Lucas Polynomials
Riordan arrays are useful for solving the combinatorial sums by the help of generating functions. Many theorems can be easily proved by Riordan arrays. In this paper we consider the Pascal matrix and define a new generalization of Fibonacci polynomials called p, q -Fibonacci polynomials. We obtain combinatorial identities and by using Riordanmethodwe get factorizations of Pascal matrix involvin...
متن کاملRiordan-Bernstein Polynomials, Hankel Transforms and Somos Sequences
Using the language of Riordan arrays, we define a notion of generalized Bernstein polynomials which are defined as elements of certain Riordan arrays. We characterize the general elements of these arrays, and examine the Hankel transform of the row sums and the first columns of these arrays. We propose conditions under which these Hankel transforms possess the Somos-4 property. We use the gener...
متن کاملTotal positivity of Riordan arrays
An infinite matrix is called totally positive if its minors of all orders are nonnegative. A nonnegative sequence (an)n≥0 is called log-convex (logconcave, resp.) if aiaj+1 ≥ ai+1aj ( aiaj+1 ≤ ai+1aj , resp.) for 0 ≤ i < j . The object of this talk is to study various positivity properties of Riordan arrays, including the total positivity of such a matrix, the log-convexity of the 0th column an...
متن کاملCombinatorial Polynomials as Moments, Hankel Transforms, and Exponential Riordan Arrays
In the case of two combinatorial polynomials, we show that they can exhibited as moments of paramaterized families of orthogonal polynomials, and hence derive their Hankel transforms. Exponential Riordan arrays are the main vehicles used for this.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Discrete Mathematics
دوره 309 شماره
صفحات -
تاریخ انتشار 2009